Improving on the Adrogué-Madias Formula

Abstract

The Adrogué-Madias (A-M) formula is correct as written, but technically, it only works when adding 1 L of an intravenous (IV) fluid. For all other volumes, the A-M algorithm gives an approximate answer, one that diverges further from the truth as the IV volume is increased. If 1 L of an IV fluid is calculated to change the serum sodium by some amount, then it was long assumed that giving a fraction of the liter would change the serum sodium by a proportional amount. We challenged that assumption and now prove that the A-M change in [sodium] ([Na]) is not scalable in a linear way. Rather, the Δ[Na] needs to be scaled in a way that accounts for the actual volume of IV fluid being given. This is accomplished by our improved version of the A-M formula in a mathematically rigorous way. Our equation accepts any IV fluid volume, eliminates the illogical infinities, and most importantly, incorporates the scaling step so that it cannot be forgotten. However, the nonlinear scaling makes it harder to obtain a desired Δ[Na]. Therefore, we reversed the equation so that clinicians can enter the desired Δ[Na], keeping the rate of sodium correction safe, and then get an answer in terms of the volume of IV fluid to infuse. The improved equation can also unify the A-M formula with the corollary A-M loss equation wherein 1 L of urine is lost. The method is to treat loss as a negative volume. Because the new equation is just as straightforward as the original formula, we believe that the improved form of A-M is ready for immediate use, alongside frequent [Na] monitoring.

Keywords: Adrogué-Madias algorithm; acid/base and electrolyte disorders; algorithm; delta [sodium]; dysnatremia; general equation; hypernatremia; hyponatremia; infinity; non-linear; scaling.

Figure 1.

Graph shows how a scaled Adrogué–Madias (A-M) formula diverges from the improved version of A-M. (A) Suppose a patient with a total body water of 42 L and a serum sodium of 111 mEq/L is going to be treated with normal saline (NS). The A-M formula yields a Δ[sodium] (Δ[Na]) of $+1$ mEq/L per liter of NS that is infused $\left(\frac{154-111}{42+1}=1\right)$. Scale this Δ[Na] as desired by multiplying by any volume of NS. The result is a straight line with a slope of one (in red). To graph the improved equation, keep the baseline parameters the same for a fair comparison with A-M. Because it can accommodate any volume of NS, not just 1 L, the improved equation appears as a curve (in blue). (B) A close-up view is shown of the A-M formula and of the improved equation for scaling volumes between 0 and 1 L. The A-M formula’s line (red) almost overlaps with the improved equation’s curve (blue). The two intersect at $x=1$ L, as expected, and then deviate farther apart.